Consider biconjugate $f^{**}$ of given nonconvex function $f$.
It is well known that $f^{**}$ is the greatest convex minorant of $f$ such that $f \geq f^{**}$.
I wonder whether $f^{**}$ is also the best approximation of $f$, that is, $\hat{f} = f^{**}$ is convex and minimise $\lVert \hat{f} - f \rVert$ (for some norm).